Fundamental group of a topological group.

It is well known that the fundamental group of a topological group connected to the route is Abelian. Suppose that $$G$$ It is a topological group connected and leaves $$Ab (G)$$ The abelianization of the topological group. $$G$$. Is there a relationship between $$pi_ {1} G$$ Y $$pi_ {1} (Ab (G))$$ ?